How to install Ubuntu to a Mac Mini i7 2011 with AMD graphics?

The default EFI and bootloader on macOS will not allow booting without modification by a third party bootloader, or an arduous manual booting process every time you restart.

Moreover, since you report that the text-based installer works, I would argue that there is no conflict with the graphics card. Overall, I would recommend using the third party option, and specifically refind.

Here is what I do step by step:

1. Prepare the Mini with macOS first.

   • Reboot the machine, hold down command-R, and restore macOS (it may say OS X).
   • Once restored, let it update fully and then reboot patient and update everything on macOS first. In your case, it will probably restore to OS X Lion. That's fine - refind works on that.
   • Upon reboot, run Disk Utility, and add only one partition to the drive for your Linux OS. This is processor-intensive and slow. Be patient.

2. Prepare the bootloader refind

   • In macOS, navigate to reind download page and download refind to your Downloads folder.
   • Reboot the mac and hold down command-R for restore.

   • Open Terminal, and run the following commands:

     cd ~/Downloads/refind-bin-0.10.4
     ./refind install
     exit
     

   • Quit Terminal and reboot the machine.

3. Install guest OS

   • Power up the machine - it will now use refind bootloader - pick the Mac OS X option with arrow keys. Once you log in, enter the CD, or the flash drive for the Linux distribution you are choosing.

   • Reboot the machine. The refind bootloader will recognize the grub bootloader. You may see more than one - choose the one with the EFI naming convention.

   • Install guest OS and be careful to select the partition correctly. This will require you to set up your partitions manually in the text based installer. There are numerous tutorials on this, but I recommend setting up a minimum of 4 partitions for boot, root, home, and var.

4. Re-run refind and remove original config and create new config

   • Reboot machine and hold down command-R.

   • Open Terminal and do the following:

     cd ~/Downloads/refind-bin-0.10.4
     ./mountesp
     cd /Volumes/ESP/EFI/refind
     rm -rf ._refind.conf
     rm -rf refind.conf
     cd ~/Downloads/refind-bin-0.10.4
     ./refind-install
     

   • Reboot the machine. Select the Linux OS distribution at the refind bootloader menu.

That's it; you are done. Other considerations below.

Alternate Step 3. Different way of booting install CD / drive / etc.

A few times, I have booted into Linux using the alt key instead of refind. However, I have only had success with this as a last resort and when refind was already preinstalled onto the macOS volume. But, it has worked more than once. Essentially, you are preserving every other step, but using the native macOS bootloader one last time before you install the Linux OS, and merely to boot the install disk. I have even done this alternate method by holding down c on the keyboard. But, it only works if every other step remains the same.

A word on partitioning:

Use your judgment on partitioning. I have three dual-boot Macs. For my 500GB drive, I keep 60GB for OS X, 128, 128, 128, and swap for three different Linux flavors. On another machine, which I use in production, I keep 468GB for the macOS volume, and keep a skinny 32GB Ubuntu-MATE partition. Each machine and purpose is different.

Why keep OS X or macOS?

Keeping a small macOS or OS X boot volume is important for stability of booting and administering the Linux OS. If Linux crashes, you can easily boot into macOS and nuke and pave the partition from Terminal. Also, I have found the distributions are more stable with the refind bootloader. I have attempted to create Linux only intel Macs but they always crashed and or involved tedious steps to boot the Linux OS. Lastly, I am pretty sure your machine's firmware is fully up to date, but maybe not. Having macOS or OS X will allow you to easily upgrade and check for firmware upgrades. I have a 2010 laptop that I run Linux on primarily and recently booted back into OS X and received a firmware update that had somehow been missed.